General Recurrence Relation Research Articles

The method of constant terms and k-colored generalized Frobenius partitions

In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let cϕk(n) denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any n≥0, cϕ2(5n+3)≡0(mod5). Since then, many scholars subsequently considered congruence properties of various k-colored generalized Frobenius partition functions, typically with a small number of colors.In 2019, Chan, Wang and Yang systematically studied arithmetic properties of CΦk(q) with 2≤k≤17 by employing the theory of modular forms, where CΦk(q) denotes the generating function of cϕk(n). We notice that many coefficients in the expressions of CΦk(q) are not integers. In this paper, we first observe that CΦk(q) is related to the constant term of a family of bivariable functions, then establish a general symmetric and recurrence relation on the coefficients of these bivariable functions. Based on this relation, we next derive many bivariable identities. By extracting and computing the constant terms of these bivariable identities, we establish the expressions of CΦk(q) with integral coefficients. As an immediate consequence, we prove some infinite families of congruences satisfied by cϕk(n), where k is allowed to grow arbitrary large.

Some Axioms and Identities of L-Moments from Logistic Distribution with Generalizations

In this paper, we derive the L-moments for some distributions, such as logistic, generalized logistic, doubly truncated logistic, and doubly truncated generalized logistic distributions. We also establish some new axioms and identities, including recurrence relations satisfied by the L-moment from the underlying derivations. In addition, we establish some new general recurrence relations satisfied by the L-moment from any distribution.

Construction of dual-generalized complex Fibonacci and Lucas quaternions

The aim of this paper is to construct dual-generalized complex Fibonacci and Lucas quaternions. It examines the properties both as dual-generalized complex number and as quaternion. Additionally, general recurrence relations, Binet's formulas, Tagiuri's (or Vajda's like), Honsberger's, d'Ocagne's, Cassini's and Catalan's identities are obtained. A series of matrix representations of these special quaternions is introduced. Finally, the multiplication of dual-generalized complex Fibonacci and Lucas quaternions are also expressed as their different matrix representations.

Analysis of Limiting Ratios of Special Sequences

In this paper, we have determined the limit of ratio of (n+1)th term to the nth term of famous sequences in mathematics like Fibonacci Sequence, Fibonacci – Like Sequence, Pell's Sequence, Generalized Fibonacci Sequence, Padovan Sequence, Generalized Padovan Sequence, Narayana Sequence, Generalized Narayana Sequence, Generalized Recurrence Relations of Fibonacci – Type sequence, Polygonal Numbers, Catalan Sequence, Cayley numbers, Harmonic Numbers and Partition Numbers. We define this ratio as limiting ratio of the corresponding sequence. Sixteen different classes of special sequences are considered in this paper and we have determined the limiting ratios for each one of them. In particular, we have shown that the limiting ratios of Fibonacci sequence and Fibonacci – Like sequence is the fascinating real number called Golden Ratio which is 1.618 approximately. We have shown that the limiting ratio of Pell's sequence is a real number called Silver Ratio and the limiting ratios for generalized Fibonacci sequence are metallic ratios. We have also obtained the limiting ratios of Padovan and generalized Padovan sequence. The limiting ratio of Narayana sequence happens to be a number called super Golden Ratio which is 1.4655 approximately. We have shown that the limiting ratios of Generalized Narayana sequence are the numbers known as super Metallic Ratios. We have also shown that the limiting ratio of generalized recurrence relation of Fibonacci type is 2 and that of Polygonal numbers and Harmonic numbers are 1. We have proved that the limiting ratio of the famous Catalan sequence and Cayley numbers are 4. Finally, assuming Rademacher's Formula, we have shown that the limiting ratio of Partition numbers is the natural logarithmic base e. We have proved fourteen theorems to derive limiting ratios of various well known sequences in this paper. From these limiting ratio values, we can understand the asymptotic behavior of the terms of all these amusing sequences of numbers in mathematics. The limiting ratio values also provide an opportunity to apply in lots of counting and practical problems.

A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences , first encountered as a special sequence (corresponding to ) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for , and we derive an exponential generating function of expressed by Kummer's confluent hypergeometric function.

Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations

By introducing for monotone recurrence relations pseudo solutions, which are analogues of pseudo orbits of dynamical systems, we show that for general monotone recurrence relations the rotation set is closed, and each element in the rotation set is realized by a Birkhoff orbit. Moreover, if there is an orbit without rotation number, then the system has positive topological entropy, and we can construct orbits shadowing different rotation numbers.

Image Reconstruction Based on Novel Sets of Generalized Orthogonal Moments.

In this work, we have presented a general framework for reconstruction of intensity images based on new sets of Generalized Fractional order of Chebyshev orthogonal Moments (GFCMs), a novel set of Fractional order orthogonal Laguerre Moments (FLMs) and Generalized Fractional order orthogonal Laguerre Moments (GFLMs). The fractional and generalized recurrence relations of fractional order Chebyshev functions are defined. The fractional and generalized fractional order Laguerre recurrence formulas are given. The new presented generalized fractional order moments are tested with the existing orthogonal moments classical Chebyshev moments, Laguerre moments, and Fractional order Chebyshev Moments (FCMs). The numerical results show that the importance of our general framework which gives a very comprehensive study on intensity image representation based GFCMs, FLMs, and GFLMs. In addition, the fractional parameters give a flexibility of studying global features of images at different positions and scales of the given moments.

Hartmann Potential with a Minimal Length and Generalized Recurrence Relations for Matrix Elements

In this work we study the Schr\"{o}dinger equation in the presence of the Hartmann potential with a generalized uncertainty principle. We pertubatively obtain the matrix elements of the hamiltonian at first order in the parameter of deformation $\beta$ and show that some degenerate states are removed. We give analytic expressions for the solutions of the diagonal matrix elements. Finally, we derive a generalized recurrence formula for the angular average values.

A matrix approach to some second-order difference equations with sign-alternating coefficients

ABSTRACTIn this paper, we analyse and unify some recent results on the double sequence , for , defined by the second-order difference equation with and , in terms of matrix theory and orthogonal polynomials theory. Moreover, we provide a general solution to using a closely related approach. We discuss briefly other recent problems involving a general recurrence relation of second order and relate them with the existing literature.

General Linear Recurrence Sequences and Their Convolution Formulas

We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to find convolution formulas for second order linear recurrence polynomials generated by 1 1 + a t + b t 2 x . The case of generating functions containing parameters, even in the numerator is considered. Convolution formulas and general recurrence relations are derived. Many illustrative examples and a straightforward extension to the case of matrix polynomials are shown.

A mixed scheme of product integration rules in (−1,1)

The paper deals with the numerical approximation of integrals of the typeI(f,y):=∫−11f(x)k(x,y)dx,y∈S⊂R where f is a smooth function and the kernel k(x,y) involves some kinds of “pathologies” (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a “mixed” quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure.

Lamb’s solution and the stress moments for a sphere in Stokes flow

Abstract This paper describes a method to generate expressions for the moments of the fluid stress integrated over a sphere in a general Stokes flow based on Lamb’s general solution of the Stokes equations. Explicit results up to moments of order four are given together with a general recurrence relation for moments of higher order. The connection of the results to earlier expressions for the stress moments leading, in particular, to generalized Faxen-type relations is demonstrated and the connection with representations of the rotation group is addressed. This study is motivated by the central role played by the Lamb coefficient in the physalis method for the resolved simulation of particulate flows. An example of the application of this method to the shear flow of a suspension of 3200 equal spheres at finite Reynolds number is given. In addition, the present theory is applied to the analysis of the particle contribution to the mixture stress in particulate flows at both finite and vanishing Reynolds numbers. It is shown that, in the case of spatial non-uniformity, the mixture stress acquires new contributions, including a new viscosity parameter, which are explicitly calculated for the case of a dilute suspension with negligible inertia.

CHARACTERIZATION OF THE SUM OF BINOMIAL RANDOM VARIABLES UNDER RANKED SET SAMPLING

Abstract In this paper, we examined the characteristics of the sum of independent and non-identical set of binomial ranked set samples, where each set has different order depending success probability. The characterization is done by establishing the general recurrence relations for two different situations based on the number of cycle, which is initially pre-assumed as a constant integer and when it is a random variable. To extend the knowledge about the characteristics of sum in terms of their behaviour and pattern, first four moments i.e., mean, variance, skewness and kurtosis are derive and compared with the sum of binomial simple random samples with same success probability. The proposed procedure has been illustrated through a real-life data on survivorship of children below one year in Empowered Action Groups (EAG) states of India.

English

This paper examines a general recurrence relation by the use of fractional reduced differential transform and then a scheme (methodology) on how to find closed solutions of one dimensional time fractional diffusion equations with initial conditions in the form of infinite fractional power series and in terms of Mittag-Leffler function in one parameter as well as their exact solutions by the use of fractional reduced differential transform method. The new general recurrence relation and the methodology of the fractional reduced differential transform method were successfully developed. The obtained new general recurrence relation helps us to solve time-fractional diffusion equations with initial conditions and various external forces by using fractional reduced differential transform method. To see its effectiveness and applicability, five test examples were presented. The results show that the general recurrence relation works successfully in solving time-fractional diffusion equations in a direct way without using linearization, transformations, perturbation, discretization or restrictive assumptions by using fractional reduced differential transform method.   Key words: Time fractional diffusion equations with initial conditions, Caputo fractional derivatives, Mittag-Leffler function, Fractional reduced differential transform method (FRDTM).

The Quantization of the E ⊗ e Jahn-Teller Hamiltonian.

The E ⊗ e Jahn-Teller Hamiltonian in the Bargmann-Fock representation gives rise to a system of two coupled first-order differential equations in the complex field, which may be rewritten in the Birkhoff standard form. General leapfrog recurrence relations are derived, from which the quantized solutions of these equations can be obtained. The results are compared to the analogous quantization scheme for the Rabi Hamiltonian.

On (p,q)-classical orthogonal polynomials and their characterization theorems

In this paper, we introduce a general (p, q)-Sturm-Liouville difference equation whose solutions are (p, q)-analogues of classical orthogonal polynomials leading to Jacobi, Laguerre, and Hermite polynomials as (p, q) to(1,1). In this direction, some basic characterization theorems for the introduced (p, q)-Sturm-Liouville difference equation, such as Rodrigues representation for the solution of this equation, a general three-term recurrence relation, and a structure relation for the (p, q)-classical polynomial solutions are given.

Exact and Approximate Solutions of Fractional Diffusion Equations with Fractional Reaction Terms

In this paper, we consider fractional reaction-diffusion e quations with linear and nonlinear fractional reaction ter ms in a semi-infinite domain. Using q-Homotopy Analysis Method, so lutions to these equations are obtained in the form of general recurrence relations. Closed form solutions in the form of the Mittag-Leffler function are perfectly obtained in the case with linea r fractional reaction term due to the ability to control the auxiliary par ameter h. Series solution is obtained for the case of nonlinear fract ional reaction term. Numerical analysis is presented for this cas e to display the fast convergent rate of the series solution o btained. q-HAM is a relatively simple and powerful method and has advantages over some other methods which we discuss and demonstrate for some initial value problems.

A family of summation formulae involving harmonic numbers

In terms of the derivative operator and a binomial sum from the telescoping method, we establish a family of summation formulae involving harmonic numbers. Meanwhile, a general recurrence relation is also deduced in accordance with Abel's lemma on summation by parts.

Recurrence Relations for Single and Product Moments of Dual Generalized Order Statistics from a General Class of Distributions

In this paper we derive some general recurrence relations between single and product moments of dual gener- alized order statistics from a general class of distributions, thus generalizing and unifying the earlier results in this direction due to several authors.

An efficient algorithm for the calculation of reserves for non-unit linked life policies

The underlying stochastic nature of the requirements for the Solvency II regulations has introduced significant challenges if the required calculations are to be performed correctly, without resorting to excessive approximations, within practical timescales. It is generally acknowledged by practising actuaries within UK life offices that it is currently impossible to correctly fulfil the requirements imposed by Solvency II using existing computational techniques based on commercially available valuation packages. Our work has already shown that it is possible to perform profitability calculations at a far higher rate than is achievable using commercial packages. One of the key factors in achieving these gains is to calculate reserves using recurrence relations that scale linearly with the number of time steps. Here, we present a general vector recurrence relation which can be used for a wide range of non-unit linked policies that are covered by Solvency II; such contracts include annuities, term assurances, and endowments. Our results suggest that by using an optimised parallel implementation of this algorithm, on an affordable hardware platform, it is possible to perform the ‘brute force’ approach to demonstrating solvency in a realistic timescale (of the order of a few hours).